Lemma 6 (Generalised fundamental identity of time-frequency analysis — see [1]}, [2]}) 

Let \(\theta \in \mathbb {R}\)  and \(f,g \in L^2(\mathbb {R})\) . It holds that
\(\mathcal {V}_{\mathcal {F}_\theta g} \mathcal {F}_\theta f (x,\omega ) = \mathcal {V}_g f (R_\theta (x,\omega )) \mathrm {e}^{\pi \mathrm {i}\sin \theta \left( \left( x^2 - \omega ^2 \right) \cos \theta - 2 x\omega \sin \theta \right)},\) 
